To write the formula for the given sinusoidal function [tex]\( f(x) \)[/tex], we need to determine its amplitude, frequency, and vertical shift based on the given points:
1. Vertical Shift (Midline Intersection):
The function intersects its midline at the point [tex]\((0, -7)\)[/tex], which indicates that the vertical shift [tex]\(D\)[/tex] is [tex]\( -7 \)[/tex].
2. Minimum Point:
The minimum point of the function is given at [tex]\(\left(\frac{\pi}{4}, -9\right)\)[/tex].
3. Amplitude:
The amplitude ([tex]\(A\)[/tex]) is the distance from the midline to either the maximum or the minimum value. Here, the midline value is [tex]\(-7\)[/tex] and the minimum value is [tex]\(-9\)[/tex], so the amplitude is:
[tex]\[
A = |-9 - (-7)| = | -2 | = 2
\][/tex]
4. Period and Frequency:
We know that the minimum occurs at [tex]\(\frac{\pi}{4}\)[/tex], and since this is one quarter of the period (the distance from the midline to the minimum), the full period [tex]\(T\)[/tex] can be derived as:
[tex]\[
\frac{T}{4} = \frac{\pi}{4} \implies T = \pi
\][/tex]
The frequency [tex]\(B\)[/tex] is related to the period by the formula:
[tex]\[
B = \frac{2\pi}{T} = \frac{2\pi}{\pi} = 2
\][/tex]
Given these parameters:
- Amplitude [tex]\(A = 2\)[/tex]
- Frequency [tex]\(B = 2\)[/tex]
- Vertical shift (midline) [tex]\(D = -7\)[/tex]
The formula for our sinusoidal function is:
[tex]\[
f(x) = 2 \sin(2x) - 7
\][/tex]
